Number theoretic properties of generating functions ...€¦ · AMS/MAA Joint Mathematics Meetings...

Number theoretic propertiesof generating functions related to

Dyson’s rankfor partitions into distinct parts.

Maria Monks

[email protected]

AMS/MAA Joint Mathematics Meetings - Washington, DC – p.1/21

Definitions

A partition λ of a positive integer n is anonincreasing sequence (λ1, λ2, . . . , λm) ofpositive integers whose sum is n. Each λi iscalled a part of λ.


Definitions


A partition into distinct parts is a partitionwhose parts are all distinct.


Definitions



p(n) is the number of partitions of n.


Definitions



p(n) is the number of partitions of n.

Q(n) is the number of partitions of n intodistinct parts.


The underlying problemSince the functions p(n) and Q(n) have no known elegant closedformula, we wish to uncover some of their number-theoreticproperties.



Ramanujan discovered the famous congruence identities

p(5n+ 4) ≡ 0 (mod 5)

p(7n+ 5) ≡ 0 (mod 7)

p(11n+ 6) ≡ 0 (mod 11)




p(5n+ 4) ≡ 0 (mod 5)

p(7n+ 5) ≡ 0 (mod 7)

p(11n+ 6) ≡ 0 (mod 11)

Similar identities have been found for Q(n). For instance,Q(5n+ 1) ≡ 0 (mod 4) whenever n is not divisible by 5.




p(5n+ 4) ≡ 0 (mod 5)

p(7n+ 5) ≡ 0 (mod 7)

p(11n+ 6) ≡ 0 (mod 11)

Similar identities have been found for Q(n). For instance,Q(5n+ 1) ≡ 0 (mod 4) whenever n is not divisible by 5.

Are there combinatorial explanations for these elegant identities?


Dyson’s rankFreeman Dyson conjectured that there is a combinatorial invariantthat sorts the partitions of 5n+ 4 into 5 equal-sized groups, thusexplaining the congruence p(5n+ 4) ≡ 0 (mod 5).



Dyson defined the rank of a partition λ = (λ1, . . . , λm) to beλ1 −m. For example, the rank of the following partition is 1:



Dyson defined the rank of a partition λ = (λ1, . . . , λm) to beλ1 −m. For example, the rank of the following partition is 1:

{{

m = 4

λ1 = 5


Combinatorial intepretations

Atkin and Swinnerton-Dyer: When thepartitions of 5n + 4 are sorted by their rankmodulo 5, the resulting 5 sets all have thesame number of elements!




Taken modulo 7, the rank also sorts thepartitions of 7n+5 into 7 equal-sized groups.




Taken modulo 7, the rank also sorts thepartitions of 7n + 5 into 7 equal-sized groups.

Failed to explain p(11n + 6) ≡ 0 (mod 11).Garvan discovered the crank, whichexplained this identity along with many othercongruences.


The rank and Q(n)

Gordon and Ono: For any positive integer j, the set of integers nfor which Q(n) is divisible by 2j is dense in the positive integers.


The rank and Q(n)


Can a rank or similar combinatorial invariant be used to explaincongruences for Q(n)?


The rank and Q(n)


Can a rank or similar combinatorial invariant be used to explaincongruences for Q(n)?

The rank provides a combinatorial interpretation for j = 1 and j = 2!

Theorem (M.). Define T (m, k;n) to be the number of partitions of n into dis-

tinct parts having rank congruent to m (mod k). Then

T (0, 4;n) = T (1, 4;n) = T (2, 4;n) = T (3, 4;n)

if and only if 24n + 1 has a prime divisor p 6≡ ±1 (mod 24) such that the

largest power of p dividing 24n+ 1 is pe where e is odd.


Outline of proofFranklin’s Involution φ:



φ



φ

The fixed points of Franklin’s Involution are the pentagonalpartitions, with k(3k ± 1)/2 squares:



φ

The fixed points of Franklin’s Involution are the pentagonalpartitions, with k(3k ± 1)/2 squares:

{k


Outline of proof

Unless n = k(3k ± 1)/2, the rank of any partition λ of n into distinctparts differs from that of φ(λ) by 2.


Outline of proof


For n 6= k(3k ± 1)/2, T (0, 4;n) = T (2, 4;n) andT (1, 4;n) = T (3, 4;n).


Outline of proof



Andrews, Dyson, Hickerson: T (0, 2;n) = T (1, 2;n) if and only if24n+ 1 has a prime divisor p 6≡ ±1 (mod 24) such that the largestpower of p dividing 24n+ 1 is pe for some odd positive integer e.


Outline of proof



Andrews, Dyson, Hickerson: T (0, 2;n) = T (1, 2;n) if and only if24n+ 1 has a prime divisor p 6≡ ±1 (mod 24) such that the largestpower of p dividing 24n+ 1 is pe for some odd positive integer e.

Thus T (0, 4;n) = T (1, 4;n) = T (2, 4;n) = T (3, 4;n) for such n, andthe set of such n is dense in the integers. Thus Q(n) is nearlyalways divisible by 4.


Generating functions

Let Q(n, r) denote the number of partitions of n intodistinct parts having rank r, and define

G(z, q) =∑

n,r

Q(n, r)zrqn.


Generating functions

Let Q(n, r) denote the number of partitions of n intodistinct parts having rank r, and define

G(z, q) =∑

n,r

Q(n, r)zrqn.

One can show that

G(z, q) = 1 +

∞∑

s=1

qs(s+1)/2

(1 − zq)(1 − zq2) · · · (1 − zqs)

for z, q ∈ C with |z| ≤ 1, |q| < 1.


G(z, q) at fourth roots of unity z

Theorem (M.). Let q ∈ C with |q| < 1. Then

G(i, q) =∞∑

k=0

ikqk(3k+1)/2 +∞∑

k=1

ik−1qk(3k−1)/2

G(−i, q) =∞∑

k=0

(−i)kqk(3k+1)/2 +∞∑

k=1

(−i)k−1qk(3k−1)/2

G(1, q) =∑∞

n=0Q(n)qn = (1 + q)(1 + q2)(1 + q3) · · · is a weight 0

modular form, in the variable τ where q = e2πiτ .


G(z, q) at fourth roots of unity z

Theorem (M.). Let q ∈ C with |q| < 1. Then

G(i, q) =∞∑

k=0

ikqk(3k+1)/2 +∞∑

k=1

ik−1qk(3k−1)/2

G(−i, q) =∞∑

k=0

(−i)kqk(3k+1)/2 +∞∑

k=1

(−i)k−1qk(3k−1)/2

G(1, q) =∑∞

n=0Q(n)qn = (1 + q)(1 + q2)(1 + q3) · · · is a weight 0

modular form, in the variable τ where q = e2πiτ .

G(−1, q) =∑∞

n=0(T (n; 0, 2) − T (n; 1, 2))qn has been studied indepth by Andrews, Dyson, and Hickerson.


A new false theta function (or two)

It follows that

qG(i, q24) =

∞∑

k=0

ikq(6k+1)2 +

∞∑

k=1

ik−1q(6k−1)2

and

qG(−i, q24) =∞∑

k=0

(−i)kq(6k+1)2 +∞∑

k=1

(−i)k−1q(6k−1)2 .


A new false theta function (or two)

It follows that

qG(i, q24) =

∞∑

k=0

ikq(6k+1)2 +

∞∑

k=1

ik−1q(6k−1)2

and

qG(−i, q24) =∞∑

k=0

(−i)kq(6k+1)2 +∞∑

k=1

(−i)k−1q(6k−1)2 .

Not true theta functions, but they resemble theta functions in thesense that their coefficients are roots of unity and are 0 wheneverthe exponent of q is not a perfect square. Such functions areknown as false theta functions.


More generating functions

Let p(n, r) denote the number of partitions of n having rank

r, and define

R(z, q) =∑

n,r

p(n, r)zrqn.




r, and define

R(z, q) =∑

n,r

p(n, r)zrqn.

One can show that

R(z, q) = 1 +

∞∑

n=1

qn2

∏nk=1(1 − zqk)(1 − z−1qk)

for z, q ∈ C with |z| ≤ 1, |q| < 1.




r, and define

R(z, q) =∑

n,r

p(n, r)zrqn.

One can show that

R(z, q) = 1 +

∞∑

n=1

qn2

∏nk=1(1 − zqk)(1 − z−1qk)

for z, q ∈ C with |z| ≤ 1, |q| < 1.

R(−1, q) is one of Ramanujan’s famous “mock theta

functions”.


The relation betweenG and R

Theorem (M.). We have

R(i, 1/q) = R(−i, 1/q) =1 − i

2G(i, q) +

1 + i

2G(−i, q)




R(i, 1/q) = R(−i, 1/q) =1 − i

2G(i, q) +

1 + i

2G(−i, q)

or alternatively,

qR(i, q−24) =∞∑

n=0

(−1)n(q(12n+1)2 + q(12n+5)2 + q(12n+7)2 + q(12n+11)2

)

= q + q25 + q49 + q121 − q169

−q289 − q361 − q529 + q625 + · · · .




R(i, 1/q) = R(−i, 1/q) =1 − i

2G(i, q) +

1 + i

2G(−i, q)

or alternatively,

qR(i, q−24) =∞∑

n=0

(−1)n(q(12n+1)2 + q(12n+5)2 + q(12n+7)2 + q(12n+11)2

)

= q + q25 + q49 + q121 − q169

−q289 − q361 − q529 + q625 + · · · .

The analytic behavior of the false theta functions G(±i, q) gives thebehavior of R(±i, q) for q outside the unit disk!


Relation to modular forms

Bringmann and Ono: If z 6= 1 is a root of unity,the function R(z, q) is the “holomorphic part”of a weight 1/2 harmonic Maass form.




Therefore, the functions G(±i, q−1) appearnaturally within the theory of automorphicforms.




Therefore, the functions G(±i, q−1) appearnaturally within the theory of automorphicforms.

What about G(w, q), and G(w, q−1), for otherroots of unity w?


Relating G(w, q) to modular forms

Define the series

D(w; q) = (1 + w)G(w; q) + (1 − w)G(−w; q).



Define the series

D(w; q) = (1 + w)G(w; q) + (1 − w)G(−w; q).

For roots of unity ζ 6= 1, the following is a weight 0 modular form.

η(ζ; τ) := q1

12

∞∏

n=1

(1 − ζqn)(1 − ζ−1qn) = q1

12 (ζq; q)∞(ζ−1q; q)∞



Define the series

D(w; q) = (1 + w)G(w; q) + (1 − w)G(−w; q).

For roots of unity ζ 6= 1, the following is a weight 0 modular form.

η(ζ; τ) := q1

12

∞∏

n=1

(1 − ζqn)(1 − ζ−1qn) = q1

12 (ζq; q)∞(ζ−1q; q)∞

Theorem (M., Ono). We have

q1

12 ·D(ζ; q)D(ζ−1; q) = 4 ·η(2τ)4

η(τ)2η(ζ2; 2τ)

is a weight 1 modular form for roots of unity ζ 6= ±1.


The function G(w, q)Define G(w, q) = G(w, q−1). Formal manipulation yields

G(w, q) =∑

n≥0

(−w−1)n

(w−1q; q)n.



G(w, q) =∑

n≥0

(−w−1)n

(w−1q; q)n.

This is not a well-defined q-series, but we can fix this by

considering Gt(w, q) =∑t

n=0(−w−1)n

(w−1q;q)n

.



G(w, q) =∑

n≥0

(−w−1)n

(w−1q; q)n.

This is not a well-defined q-series, but we can fix this by

considering Gt(w, q) =∑t

n=0(−w−1)n

(w−1q;q)n

.

Theorem (M., Ono). Suppose that −w−1 6= 1 is anmth primitive root of unity.

If 0 ≤ r < m, then limn→∞ Gmn+r(w; q) is a well defined q-series and

satisfies

limn→∞

Gmn+r(w; q) = limn→∞

Gmn(w; q) +(−w−1)r − 1

w + 1

1

(w−1q; q)∞.


Example: The case−w−1 = −1

G1(1; q) = −q − q2 − q3 − q4 − q5 − q6 − q7 − q8 − q9 − · · ·

G3(1; q) = −q − q2 − 2q3 − 2q4 − 3q5 − 4q6 − 5q7 − 6q8 − 8q9 − · · ·

G5(1; q) = −q − q2 − 2q3 − 2q4 − 4q5 − 5q6 − 7q7 − 9q8 − 13q9 − · · ·

G7(1; q) = −q − q2 − 2q3 − 2q4 − 4q5 − 5q6 − 8q7 − 10q8 − 15q9 − · · ·

G9(1; q) = −q − q2 − 2q3 − 2q4 − 4q5 − 5q6 − 8q7 − 10q8 − 16q9 − · · ·

and

G2(1; q) = 1 + q2 + q3 + 2q4 + 2q5 + 3q6 + 3q7 + 4q8 + 4q9 + · · ·

G4(1; q) = 1 + q2 + q3 + 3q4 + 3q5 + 5q6 + 6q7 + 9q8 + 10q9 + · · ·

G6(1; q) = 1 + q2 + q3 + 3q4 + 3q5 + 6q6 + 7q7 + 11q8 + 13q9 + · · ·

G8(1; q) = 1 + q2 + q3 + 3q4 + 3q5 + 6q6 + 7q7 + 12q8 + 14q9 + · · ·



For −w−1 a primitive mth root of unity, it now makes sense todefine the q-series

G(w, q) = limn→∞

Gmn(w; q).





Gmn(w; q).

Consider a twist of the third-order mock theta function ofRamanujan:

ψ(w, q) :=∑

n≥0

qn2

wn

(q; q2)n.





Gmn(w; q).

Consider a twist of the third-order mock theta function ofRamanujan:

ψ(w, q) :=∑

n≥0

qn2

wn

(q; q2)n.

Also define D(w, q) = (1+w−1)G(w, q)+ (1−w−2)(ψ(−w2, q)− 1).



Theorem (Folsom). Let −ω−1 6= 1 be a primitive mth root of unity. Then

q−1/12D(ω, q)D(ω−1, q) is the weight 1 modular form

q−1/12D(ω, q)D(ω−1q) =η4(q2)η2(ω2, q)

η2(q)η3(ω2, q2)

where η(ω, q) = q1/12(ωq; q)∞(ω−1q; q)∞.



Theorem (Folsom). Let −ω−1 6= 1 be a primitive mth root of unity. Then

q−1/12D(ω, q)D(ω−1, q) is the weight 1 modular form

q−1/12D(ω, q)D(ω−1q) =η4(q2)η2(ω2, q)

η2(q)η3(ω2, q2)

where η(ω, q) = q1/12(ωq; q)∞(ω−1q; q)∞.

Thus G and G appear naturally within the theory of automorphicforms!


Observations and Future WorkThe rank fails to explain the divisibility of Q(n) by higher

powers of 2. Is there a generalization of the rank that can be

used to divide the partitions of Q(n) into m equal-sized

groups whenever Q(n) is divisible by m for any positive

integer m?






integer m?

Are there other partition functions for which we can obtain

congruences via the rank or related combinatorial

invariants?






integer m?

Are there other partition functions for which we can obtain

congruences via the rank or related combinatorial

invariants?

We have seen that G(z, q) and R(z, q) are related at z = ±i.

Are these the only values of z for which they are related in

some way?


Acknowledgments

This research was done at the University of Minnesota Duluth withthe financial support of the National Science Foundation andDepartment of Defense (grant number DMS 0754106) and theNational Security Agency (grant number H98230-06-1-0013).


Acknowledgments


This work was also supported by Ken Ono’s NSF Director’sDistinguished Scholar Award, which supported my visit to theUniversity of Wisconsin in July 2008.


Acknowledgments


This work was also supported by Ken Ono’s NSF Director’sDistinguished Scholar Award, which supported my visit to theUniversity of Wisconsin in July 2008.

Thanks to Joe Gallian, Nathan Kaplan, and Ricky Liu for theirmentorship and support throughout this research project, and toKen Ono for his helpful insights and direction.


Number theoretic properties of generating functions ...€¦ · AMS/MAA Joint Mathematics Meetings...

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